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356 XI ELEMENTARY SPECTRAL THEORY

Uft \l/2

x'_n² dt\ , lim y_n = + oo; consider
I / /1-+00

the sequence of the functions y_n = x_n/y_n, and derive a contradiction from the fact that,

f*
on one hand lim yl dt — 0, and on the other hand, there is an interval [a, c] <= I and

n-»-ao Ja

a number p > 0 such that |j>_n(0| ^ p for every « and every point t e [a, c].)

(d) Let JLCI be the g.l.b. of 3>(x) in A n H₀. Show that if (x_n) is a sequence in A n H₀such that lim ®(x_tt) = ^_1} (*,,) is bounded in H (same method as in (c)). Deduce from

n-n»

that result that, by extracting a convenient subsequence, one may assume that the
sequence (x_n) is uniformly convergent in I to a function u (which, however, need not
a priori belong to H) (use Ascoli's theorem (7.5.7)).

(e) $(*) is a quadratic form in H₀, i.e. one has ®(.x + y) = 0>(x) + ®(y) -f 2 T(x, y),
where T is bilinear; for any function z which is twice continuously differentiate in I

and such that z^f(a) = z'(b) = z(a) = z(b) = 0, one has ^(x, z) = - P xz" dt H- P <?xz dt;

Yfa, z) can be defined by the same formula for any function v continuous in I. Show
that for any such function z and any real number £, one has

lim ($(*„ + |z) / f \x_n + £z)² dt]

-*oo \ / Ja I

and deduce from that result that one must have

C^b

(uz" - quz -f p&z) dt = 0.

Hence, if w is a twice continuously difTerentiable function such that w" = qu — p,^,

/*&
one has (u — w)z" dt — 0 by integration by parts ; conclude that u — w is a polynomial

of degree ^ 1 (observe that by substracting from u — w a suitable polynomial p of degree
1, there exists a function z such that z" = u — w — p, z(a) = z(b) = z'(d) = z'(b) = 0),
Hence u is twice continuously difTerentiable, satisfies the differential equation

u" — qu + pin = 0,

C^band is such that u² dt — 1 ; furthermore, u'(a) = — aw(a), w'(^) = fiu(b}. (To prove the

last statement, express that for any z e H₀ , <&(u + |z) ^ jUi (w -f ^z)² dt, for any real

number £.)

2. (a) With the notations of (11 .7.10), suppose first that k^k₂ =£ 0, and let a = hi/k_l9jg =: —h^\k^ . Show that the <p_n can be defined (up to sign) by the following conditions :
(1) 9! is such that, on the sphere A: (y \ y) = 1 in G, the function <D (defined in Problem
l(c)) reaches its minimum for y = <p_lt and that minimum is equal to A₄; (2) for n > 1,
let A_n be the intersection of A and of the hyperplanes (y \ 9?_fc) = 0 for 1 < k ^ n — 1 ;
then y>_n is such that on A_w , 0 reaches its minimum for y = <p_n , and that minimum is
equal to X_n . (The characterization of <pi follows at once from the results of Problem 1 ;
use the same kind of argument to characterize <p_n •)

(b) If ki = 0, &₂ ^ 0, prove similar results, replacing a by 0 in <D, but replacing the
sphere A by its intersection with the hyperplane in G defined by y(a) = 0. Proceed
similarly when k± ^ 0 and k₂ = 0, or when Ar_x = k₂ = 0.



	356 XI ELEMENTARY SPECTRAL THEORY Uft \l/2 x'_n² dt\ , lim y_n = + oo; consider I / /1-+00 the sequence of the functions y_n = x_n/y_n, and derive a contradiction from the fact that, f* on one hand lim yl dt — 0, and on the other hand, there is an interval [a, c] <= I and n-»-ao Ja a number p > 0 such that \|j>_n(0\| ^ p for every « and every point t e [a, c].) (d) Let JLCI be the g.l.b. of 3>(x) in A n H₀. Show that if (x_n) is a sequence in A n H₀such that lim ®(x_tt) = ^_1} (,,) is bounded in H (same method as in (c)). Deduce from n-n» that result that, by extracting a convenient subsequence, one may assume that the sequence (x_n)* is uniformly convergent in I to a function u (which, however, need not a priori belong to H) (use Ascoli's theorem (7.5.7)). (e) $() is a quadratic form in H₀, i.e. one has ®(.x + y) = 0>(x) + ®(y)* -f 2 T(x, y), where T is bilinear; for any function z which is twice continuously differentiate in I and such that z^f(a) = z'(b) = z(a) = z(b) = 0, one has ^(x, z) = - P xz" dt H- P <?xz dt; Yfa, z) can be defined by the same formula for any function v continuous in I. Show that for any such function z and any real number £, one has

	lim ($(„ + \|z) / f \x_n +* £z)² dt] -oo \ / Ja* I

	and deduce from that result that one must have C^b (uz" - quz -f p&z) dt = 0.

	Hence, if w is a twice continuously difTerentiable function such that w" = qu — p,^, /& one has (u — w)z" dt —* 0 by integration by parts ; conclude that u — w is a polynomial Ja of degree ^ 1 (observe that by substracting from u — w a suitable polynomial p of degree 1, there exists a function z such that z" = u — w — p, z(a) = z(b) = z'(d) = z'(b) = 0), Hence u is twice continuously difTerentiable, satisfies the differential equation

	u" — qu + pin = 0, C^band is such that u² dt — 1 ; furthermore, u'(a) = — aw(a), w'(^) = fiu(b}. (To prove the Ja r* last statement, express that for any z e H₀ , <&(u + \|z) ^ jUi (w -f ^z)² dt, for any real Ja number £.) 2. (a) With the notations of (11 .7.10), suppose first that k^k₂ =£ 0, and let a = hi/k_l9jg =: —h^\k^ . Show that the <p_n can be defined (up to sign) by the following conditions : (1) 9! is such that, on the sphere A: (y \ y) = 1 in G, the function <D (defined in Problem l(c)) reaches its minimum for y = <p_lt and that minimum is equal to A₄; (2) for n > 1, let A_n be the intersection of A and of the hyperplanes (y \ 9?_fc) = 0 for 1 < k ^ n — 1 ; then y>_n is such that on A_w , 0 reaches its minimum for y = <p_n , and that minimum is equal to X_n . (The characterization of <pi follows at once from the results of Problem 1 ; use the same kind of argument to characterize <p_n •) (b) If ki = 0, &₂ ^ 0, prove similar results, replacing a by 0 in <D, but replacing the sphere A by its intersection with the hyperplane in G defined by y(a) = 0. Proceed similarly when k± ^ 0 and k₂ = 0, or when Ar_x = k₂ = 0.